Add mirror modifier to linked object
You cannot add modifiers to linked objects in Blender.

Although there is a solution to your problem.

First Step

Select the object and press CTRL+G to add it to a Group.

You can give the group a name by pressing the F6 key and enter a new name.

Save the file and then open a new file to go on with step 2.



Second Step

Link the group into the new file. 

Linking the group is similar to linking an object, but you go inside the Group folder and select the group name instead.

Third Step

Select the group instance that was added from previous step and press the space key. Then type Make Proxy and select the name of the object that you want to be mirrored.



Fourth Step

Now select the object whose name ends in _proxy and add a mirror modifier.

Summary

Your mesh is still linked and any updates you make in the original file will get updated in the current file where you linked.